I was thinking of a truck wheel patch, but I'd like to be sure there won't be any stains, and that the glue won't react with the detergent, for example. I don't know if it's possible.
In that case I think you want something that will 'vulcanize' with the material in the gasket, rather than being adhered... I don't know if that's feasible for your exact material but I know there are methods of rubber patching that are more akin to welding than glueing.
Or an air mattress patch, or a pool liner patch, or a jetski seat repair patch...